Full text of DOI:10.1111/0022-1082.00366

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The Journal of Finance
Discussion
ZHENYU WANG*
THE EQUITY PREMIUM IN THE UNITED STATES has been puzzling and challenging
both to theorists and empiricists in the profession of finance and economics.
Estimating the equity premium from the time series of excess equity returns
is as challenging as explaining it with economic models. The paper by Pástor
and Stambaugh deals with the estimation rather than with the economic
explanation of the equity premium. The most interesting aspect of the paper
is its employment of economists’ ideas alongside statisticians’ procedures.
The first idea is that the equity premium should be linked to volatility. The
second is that a rise in price should reduce the equity premium. The third is
that the shifts in the equity premium over regimes should not be very large.
The paper constructs a prior distribution to express those thoughts and con-
ducts Bayesian inference using MCMC methods and a model incorporating
structure-breaks. The paper’s approach is attractive because it incorporates
economic theory and intuition into the prior distribution. It is this aspect
that distinguishes economists from statisticians.
The estimate of the equity premium delivered in Pástor and Stam-
baugh’s paper is sensible. To appreciate the equity premium estimated by
Pástor and Stambaugh, in Figure 1, I plot the 30-year moving average of
equity returns, short-term interest rates, and excess returns. The data,
kindly provided by Pástor and Stambaugh, are the same as those used in
their paper except the data before 1834 are also used. The equity premium
estimated by Pástor and Stambaugh fluctuates within the range of four to
six percent. This result does not seem to disagree with the long-run real
return of seven percent that Siegel ~1998, 1999! has argued for and the
long-run average real interest rate of around two percent. It is also consis-
tent with the long-run average excess return of 4.74 percent and the asso-
ciated standard error of 1.11 percent. In addition, the estimated equity
premium rises before World War II and declines thereafter. The rise before
World War II coincides with a gradual increase in the equity return and a
downward trend in the short-term interest rate. The decline after World
War II coincides with a large increase in the equity return until the early
1960s and a steep climb of the interest rate during the period from 1960 to
1990. Finally, the estimated equity premium drops sharply during the 1990s.
This result is consistent with the views of many economists including Sie-
gel ~1999! and Shiller ~2000! who pay attention to the sky-high price-
earning ratio near the end of the 1990s.
* Columbia University, Graduate School of Business. I thank Darrel Duffie, Ravi Jagan-
nathan, and Michael Johannes for helpful comments on my discussion.

The Equity Premium and Structural Breaks
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Figure 1. 30-year moving average of equity returns and interest rates.
I. Comments
Since the main contribution of Pástor and Stambaugh’s paper is to incor-
porate economic theory and intuition into the prior distribution for this prob-
lem, I only comment on the three prior beliefs they introduced.
A. The Positive Link Between the Premium and Volatility
The prior belief in the positive link between the equity premium and vol-
atility comes from classic economic theory. In stable regime i, the prior dis-
tribution of volatility s is restricted by the premium mi as in the following
i
equation:
mi ϭ gci si2, i ϭ 1,{{{, K ϩ 1
~1!
where g is a parameter independent of i, and ci is a parameter depending on
i. The prior distributions of g and ci are gamma distributions. There is a
restriction on mi and s because the prior mean of ci is 1 and the prior
i
variance of ci is 0.2. The volatility sj, jϩ1 in transition regimes and the prob-
ability ~ pkk, k ϭ 1,{{{,2K! of breaks are independent of the equity premium
in the prior distribution. This prior belief affects both the range and shape
of the curve of the estimated equity premium, as demonstrated in Figure 4
of Pástor and Stambaugh’s paper.

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In the above mathematical setup of the prior distribution, the premium
only depends on the volatility in stable regimes. It implies that shifts in the
equity premium are driven by shifts in the volatility between stable regimes
but are not affected by the changes in the probability and magnitude of
regime shifts or the volatility in transition regimes. This structure ignores
investors’ desire to hedge the risk of regime change. At any point in time, the
possibility of experiencing a regime switch is an important part of the risk
investors have to bear, so they will discount security prices because of this
risk. When the risk of regime change and the associated premium varies,
the equity premium in a regime, therefore, should also depend on the prob-
ability ~ pkk, k ϭ 1,{{{,2K! and magnitude ~mjϩ1 Ϫ mj! of regime shifts and
the volatility ~sj, jϩ1! in transition regimes, as well as the volatility ~s ! in
i
stable regimes. Even when the risk of regime change and the associated
premium is constant over time, it is still not reflected in the model specified
in equation ~1! because the premium is assumed to be proportional to the
volatility. In the case where regime-change risk is constant over time, the
equity premium should be a linear function of volatility with an intercept,
that is, mi ϭ go ϩ gsi2, where go represents the premium on the risk of
regime change.
The premium on the risk of regime change outlined above corresponds to
the jump-risk premium in the literature of continuous-time finance. If the
jump risk can be diversified away in a portfolio consisting of a large number
of stocks, there will be no premium on this risk and the structure in equa-
tion ~1! will be correct. Since Pástor and Stambaugh assume a regime-
switching model for the aggregate equity index, the jump risk cannot be
diversified away, and there must be a premium on such risk. The evidence of
jump-risk premium has been shown in the early literature by Jarrow and
Rosenfeld ~1984! and more recently by Pan ~2000!. A prior distribution that
incorporates an equilibrium model of the jump-risk premium would be
interesting.
B. The Negative Relation Between the Premium and Price
The prior belief in the negative relation between the equity premium and
stock price comes from a basic accounting rule. When a security’s price moves
up, the expected return is lower if the expected cash flow from the security
and the risk-free rate do not change. The equity premium then should de-
crease before it increases. For this reason, Pástor and Stambaugh model the
equity premium during transition regime j as
mj ϩ mjϩ1
ϩ bj~mjϩ1 Ϫ mj !,
~2!
2
where mj and mjϩ1 are, respectively, the equity premiums before and after
the transition regime j. Since the negative relation between the equity pre-

The Equity Premium and Structural Breaks
1243
mium and the stock price is captured by a negative value of the parameter
bj, they choose the prior distribution such that
1
b ; N Ϫ15.13,
ϫ 15.13 .
~3!
ͩ
ͪ
j
3
Almost all the mass of the prior probability distribution of bj is below zero.
This prior belief mainly effects the direction of the changes in the esti-
mated equity premium, especially after 1930, as demonstrated in Figure 5 of
Pástor and Stambaugh’s paper. If there are transition regimes in which the
prior belief expressed by ~2! and ~3! is imposed, a large movement of the
excess return always produces an opposite movement of the estimated eq-
uity premium. If there are no such transition regimes, the curve of the es-
timated equity premium resembles the moving average of the excess returns
plotted in Figure 1.
The assumed distribution of excess returns during transition regimes does
not model the effects of changes in the risk-free rate or changes in either the
expectation or risk of equity cash flows. A shift in the premium due to such
changes is not necessarily accompanied by a price move in the opposite di-
rection. This is a limitation of the above prior specification. If the equity
premium increases because the risk-free rate drops, it does not have to de-
crease before it increases. Neither should the equity premium decrease be-
fore it increases if it becomes higher because earnings grow faster and riskier
while the equity price remains the same. An economist who has the prior
distribution specified by equations ~2! and ~3! views the price movement as
the dominant force that changes the equity premium.
It is well known that the rate of return on short-term U.S. government
securities has changed substantially in the past. In Figure 1, the moving-
average of the risk-free rate decreases before 1960 and sharply increases
during 1960 to 1990. Changes in the risk-free rate also affect equity returns
because the lower ~higher! cost of bonds increases ~decreases! the return to
equity holders. In fact, the declining trend of the risk-free rate before 1960
is associated with the upward trend of the equity return. The rising trend of
the risk-free rate during 1960 to 1990 is associated with the downward trend
of the equity return.
It is also well known that the structure of equity cash flows has changed
during the past century and fluctuates over the business cycles. Siegel ~1998!
reports that the median earnings yield is 7.35 percent during 1871 to 1945
and 6.96 percent during 1946 to 1996. The real earnings growth is 0.72
percent per year during 1871 to 1945 and 3.25 percent during 1946 to 1996.
The median dividend yield is 5.16 percent during 1871 to 1945 and 3.75
percent during 1946 to 1996. Earnings growth is rather high in the late
1990s, but it cannot keep up with the sharp increase in the equity price.
These changes in the structure of equity cash flows might be associated with
the structural breaks of the equity premium.

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Large movements of equity prices are often associated with the changes in
fundamental economic variables such as interest rates, corporate earnings,
and dividends. Therefore, to a large extent, Pástor and Stambaugh’s prior
distribution captures the changes in the equity premium, especially the drop
of equity premium in the late 1990s, in a rather reasonable way. However, it
might be more fruitful to incorporate valuation models into the prior distri-
bution rather than the simple accounting rule. Such a prior distribution will
combine the information obtained from the interest rates and price-earnings
ratios. Given the large quantity of literature on the valuation of the aggre-
gate equity market based on fundamental economic variables, it is natural
that these variables affect economists’ prior beliefs in the equity premium.
C. The Range of Shifts in the Equity Premium
Pástor and Stambaugh believe that large shifts in the equity premium are
unlikely. To express such a belief, the prior distribution of the equity pre-
miums in stable regimes, denoted by the vector m, is specified as
1
2sm2
p~m6mT ! @ exp Ϫ
~m Ϫ mT i!'~m Ϫ mT i! , m Ͼ 0
~4!
ͫ
ͬ
where mT is a positive hyperparameter following a noninformative prior dis-
tribution, and i is the vector of 1s. Pástor and Stambaugh use the variance
sm2 to control the strength of the belief. As expected, the main effect of this
prior belief is to dampen the fluctuations of the estimated equity premium
over regimes. Comparing the solid curves in Figure 2 and Figure 4 of Pástor
and Stambaugh’s paper, the plots of the estimated equity premium are very
similar, except that the one in Figure 2 has a much narrower range than the
one in Figure 4. Without the prior belief expressed by equation ~4!, the es-
timated equity premium ranges from 4 to 10 percent rather than from 4 to
6 percent. Clearly, the prior belief expressed by equation ~4! is needed to
shrink the estimate to the desired range.
All of us should wonder why large shifts in the equity premium are un-
likely. Although Pástor and Stambaugh pose this belief as a wise man’s wis-
dom, it is useful for us to think of some economic reasons. If the equity
premium is the premium on risk, it will have large shifts either because
equity risk jumps or because investors’ risk aversion changes drastically.
Since the volatility of the U.S. equity index fluctuates within some range,
the link between the premium and the volatility should constrain the shifts
in the equity premium. The degree of risk aversion is usually implied in the
valuation model. Therefore, the prior belief in the link between the premium
and volatility and the prior belief in the valuation model should already
impose restrictions on the shifts in the equity premium. If the prior beliefs
in the two economic models work well, we should not need the informative
prior distribution expressed by equation ~4!.

The Equity Premium and Structural Breaks
1245
II. Summary
To deliver a reasonable estimate of the equity premium, Pástor and Stam-
baugh incorporate a pricing model that links the premium to volatility in
stable regimes, an accounting rule that relates the premium to price changes,
and the view that changes in the equity premium are unlikely to be extreme.
Their work points to an important and promising direction for research on
the equity premium. I suggest two extensions. First, the pricing model should
be extended to link the premium to both the risk of regime change and the
volatility in stable regimes. Second, a valuation model that uses fundamen-
tal economic variables should replace the accounting rule. I think a prior
distribution incorporating both the pricing model and the valuation model
should deliver a reasonable estimate of the equity premium. The view that
shifts in the equity premium are unlikely to be extreme should be a conse-
quence of the belief in the two economic models.
REFERENCES
Jarrow, Robert A., and Eric R. Rosenfeld, 1984, Jump risks and the intertemporal capital asset
pricing model, Journal of Business 57, 337–351.
Pan, Jun, 2000, Integrated time-series analysis of spot and option prices, Working paper, MIT.
Shiller, Robert, 2000, Irrational Exuberance ~Princeton University Press, Princeton, NJ!.
Siegel, Jeremy J., 1998, Stocks for the Long Run ~McGraw-Hill, New York!.
Siegel, Jeremy J., 1999, The shrinking equity premium, Journal of Portfolio Management 26~1!,
10–16.